1. Convex Hull
1/1/ :28 AM
Convex Hull
obstacle
start
end

2. Outline and Reading Convex hull (§12.5.2) Orientation (§12.5.1-2)
1/1/ :28 AM
Outline and Reading
Convex hull (§12.5.2)
Orientation (§ )
Sorting by angle (§12.5.5)
Graham scan (§12.5.5)
Analysis (§12.5.5)

3. Convex Hull
1/1/ :28 AM
Convex Polygon
A convex polygon is a nonintersecting polygon whose internal angles are all convex (i.e., less than p)
In a convex polygon, a segment joining two vertices of the polygon lies entirely inside the polygon
convex
nonconvex

4. Convex Hull
1/1/ :28 AM
Convex Hull
The convex hull of a set of points is the smallest convex polygon containing the points
Think of a rubber band snapping around the points

5. Special Cases The convex hull is a segment The convex hull is a point
1/1/ :28 AM
Special Cases
The convex hull is a segment
Two points
All the points are collinear
The convex hull is a point
there is one point
All the points are coincident

6. Applications Motion planning Geometric algorithms obstacle start end
Convex Hull
1/1/ :28 AM
Applications
Motion planning
Find an optimal route that avoids obstacles for a robot
Geometric algorithms
Convex hull is like a two-dimensional sorting
obstacle
start
end

7. Computing the Convex Hull
1/1/ :28 AM
Computing the Convex Hull
The following method computes the convex hull of a set of points
Phase 1: Find the lowest point (anchor point)
Phase 2: Form a nonintersecting polygon by sorting the points counterclockwise around the anchor point
Phase 3: While the polygon has a nonconvex vertex, remove it

8. Orientation b c CW a c b CCW a c b COLL a
Convex Hull
1/1/ :28 AM
Orientation
The orientation of three points in the plane is clockwise, counterclockwise, or collinear
orientation(a, b, c)
clockwise (CW, right turn)
counterclockwise (CCW, left turn)
collinear (COLL, no turn)
The orientation of three points is characterized by the sign of the determinant D(a, b, c), whose absolute value is twice the area of the triangle with vertices a, b and c a b c CW c b
CCW a c b a
COLL

9. Convex Hull
1/1/ :28 AM
Sorting by Angle
Computing angles from coordinates is complex and leads to numerical inaccuracy
We can sort a set of points by angle with respect to the anchor point a using a comparator based on the orientation function
b < c  orientation(a, b, c) = CCW
b = c  orientation(a, b, c) = COLL
b > c  orientation(a, b, c) = CW
CCW
COLL CW c c b b b c a a a

10. Removing Nonconvex Vertices
Convex Hull
1/1/ :28 AM
Removing Nonconvex Vertices
Testing whether a vertex is convex can be done using the orientation function
Let p, q and r be three consecutive vertices of a polygon, in counterclockwise order
q convex  orientation(p, q, r) = CCW
q nonconvex  orientation(p, q, r) = CW or COLL r r q q p p

11. Graham Scan p q r H p q r H p q r H for each vertex r of the polygon
Convex Hull
1/1/ :28 AM
Graham Scan
The Graham scan is a systematic procedure for removing nonconvex vertices from a polygon
The polygon is traversed counterclockwise and a sequence H of vertices is maintained
for each vertex r of the polygon
Let q and p be the last and second last vertex of H
while orientation(p, q, r) = CW or COLL remove q from H q  p p  vertex preceding p in H
Add r to the end of H p q r H p q r H p q r H

12. Convex Hull
1/1/ :28 AM
Analysis
Computing the convex hull of a set of points takes O(n log n) time
Finding the anchor point takes O(n) time
Sorting the points counterclockwise around the anchor point takes O(n log n) time
Use the orientation comparator and any sorting algorithm that runs in O(n log n) time (e.g., heap-sort or merge-sort)
The Graham scan takes O(n) time
Each point is inserted once in sequence H
Each vertex is removed at most once from sequence H
See pages for a Java implementation of this algorithm.

13. Incremental Convex Hull
1/1/ :28 AM
Incremental Convex Hull q w u e z t

14. Outline and Reading Point location Incremental convex hull Problem
1/1/ :28 AM
Outline and Reading
Point location
Problem
Data structure
Incremental convex hull
Insertion algorithm
Analysis

15. Convex Hull
1/1/ :28 AM
Point Location TH
Given a convex polygon P, a point location query locate(q) determines whether a query point q is inside (IN), outside (OUT), or on the boundary (ON) of P
An efficient data structure for point location stores the top and bottom chains of P in two binary search trees, TL and TH of logarithmic height
An internal node stores a pair (x (v), v) where v is a vertex and x (v) is its x-coordinate
An external node represents an edge or an empty half-plane P TL

16. Point Location (cont.) TH eH P q vL TL
Convex Hull
1/1/ :28 AM
Point Location (cont.) TH
To perform locate(q), we search for x(q) in TL and TH to find
Edge eL or vertex vL on the lower chain of P whose horizontal span includes x(q)
Edge eH or vertex vH on the upper chain of P whose horizontal span includes x(q)
We consider four cases
If no such edges/vertices exist, we return OUT
Else if q is on eL (vL) or on eH (vH), we return ON
Else if q is above eL (vL) and below eH (vH), we return IN
Else, we return OUT eH P q vL TL

17. Incremental Convex Hull
1/1/ :28 AM
Incremental Convex Hull
The incremental convex hull problem consists of performing a series of the following operations on a set S of points
locate(q): determines if query point q is inside, outside or on the convex hull of S
insert(q): inserts a new point q into S
hull(): returns the convex hull of S
Incremental convex hull data structure
We store the points of the convex hull and discard the other points
We store the hull points in two red-black trees
TL for the lower hull
TH for the upper hull

18. Convex Hull
1/1/ :28 AM
Insertion of a Point
In operation insert(q), we consider four cases that depend on the location of point q
A IN or ON: no change
B OUT and above: add q to the upper hull
C OUT and below: add q to the lower hull
D OUT and left or right: add q to the lower and upper hull A C D

19. Insertion of a Point (cont.)
Convex Hull
1/1/ :28 AM
Insertion of a Point (cont.) q
Algorithm to add a vertex q to the upper hull chain in Case B (boundary conditions omitted for simplicity)
We find the edge e (vertex v) whose horizontal span includes q
w  left endpoint (neighbor) of e (v)
z  left neighbor of w
While orientation(q, w, z) = CW or COLL
We remove vertex w
w  z
u  right endpoint (neighbor) of e (v)
t  right neighbor of u
While orientation(t, u, q) = CW or COLL
We remove vertex u
u  t
We add vertex q w e u z t q w u z t

20. Analysis Let n be the current size of the convex hull
1/1/ :28 AM
Analysis
Let n be the current size of the convex hull
Operation locate takes O(log n) time
Operation insert takes O((1 + k)log n) time, where k is the number of vertices removed
Operation hull takes O(n) time
The amortized running time of operation insert is O(log n)

21. Data Structure for MST Algorithm
Convex Hull
1/1/ :28 AM
Data Structure for MST Algorithm
current_best$
candidate$
next-
node b a IS
wait ∞ 9 f 3
yes 6 e 4 d 7 8 c 2 no
PEs
mask$
node$ a$ b$
parent$
root c$ d$ e$ f$

22. Quickhull Algorithm for ASC
Convex Hull
1/1/ :28 AM
Quickhull Algorithm for ASC
Reference:
[Maher, Baker, Akl, “An Associative Implementation of Classical Convex Hull Algorithms” ]
Review of Sequential Quickhull Algorithm
Suffices to find the upper convex hull of points that are on or above the line
Select point h so that the area of triangle weh is maximal.
Proceed recursively with the sets of points on or above the lines and h e w

23. Previous Illustration
Convex Hull
1/1/ :28 AM
Previous Illustration w e h

24. Example for Data Structure
Convex Hull
1/1/ :28 AM
Example for Data Structure
p1, w p7 p2
P3, e p4 p5
P6, h

25. Data Structure for Preceding Example
Convex Hull
1/1/ :28 AM
Data Structure for Preceding Example 1 p3 p1 6 2 p7 9 8 p6
ctr 7 11 p5 h 4 p4 12 IS p2 3
job$
hull$
right-pt$
area$
name$
left-point$
x-coord$
y-coord$
point$ w e PE
mask

26. ASC Quickhull Algorithm (Upper Convex Hull)
1/1/ :28 AM
ASC Quickhull Algorithm (Upper Convex Hull)
ASC-Quickhull( planar-point-set )
Initialize: ctr = 1, area$ = 0, hull$ = 0
Find the PE with the minimal x-coord$ and let w be its point$
Set its hull$ value to 1
Find the PE with the PE with maximal x-coord$ and let e be its point$
Set its hull$ to 1
All PEs set their left-pt to w and right-pt to e.
If the point$ for a PE lies above the line
Then set its job$ value to 1
Else set its job$ value to 0

27. ASC Quickhull Algorithm (cont)
Convex Hull
1/1/ :28 AM
ASC Quickhull Algorithm (cont)
Loop while parallel job$ contains a nonzero value
The IS makes its active cell those with a maximal job$ value.
Each (active) PE computes and stores the area of triangle (left-pt$, right-pt$, point$ ) in area$
Find the PE with the maximal area$ and let h be its point.
Set its hull$ value to 1
Each PE whose point$ is above
sets its job$ value to ++ctr
sets its job$ to ++ctr
Each PE with job$ < ctr -2 sets its job$ value to 0

28. Performance of ASC-Quickhull
Convex Hull
1/1/ :28 AM
Performance of ASC-Quickhull  4 6 5 1 2 3

29. Performance of ASC-Quickhull (cont)
Convex Hull
1/1/ :28 AM
Performance of ASC-Quickhull (cont)
Average Case:
Assume
Roughly 1/3 of the points above each line being processed are eliminated.
O(lg n) points are on the convex hull.
Shown to be true for randomly generated points
Then the average running time is O(lg n)
The average cost is O(n lg n)
Worst Case:
Running time is O(n).
Cost is O(n2)
Definition of cost is (running time)  (nr. of processors)

30. MASC Quickhull Algorithm
Convex Hull
1/1/ :28 AM
MASC Quickhull Algorithm
Algorithm:
Use IS1 to execute the first loop of ASC-Quickhull
When an IS completes computing the loop in ASC-Quickhull,
Idle ISs request problems from busy ISs who have inactive jobs on their job$ list.
Control of the PEs for an inactive job is transferred to the idle IS. The control of these PEs is returned to original IS after the job is finished.

31. ASC Quickhull Algorithm (cont)
Convex Hull
1/1/ :28 AM
ASC Quickhull Algorithm (cont)  1 2

32. Analysis for MASC Quickhull
Convex Hull
1/1/ :28 AM
Analysis for MASC Quickhull
Average Case:
Assumptions:
roughly 1/3 of the points above each line being processed are eliminated.
O(lg n) Instruction Streams are available.
There are O(lg n) convex hull points
The average running time is O(lg lg n)
Essentially constant time for real world problems.
Worst Case
O(n)

33. MASC Quickhull for a Limited Number of ISs
Convex Hull
1/1/ :28 AM
MASC Quickhull for a Limited Number of ISs
A manager IS is used to control the interactions of the ISs and the task workpool.
The manager assigns IS1 to execute the first loop of ASC-Quickhull
When an IS completes the execution of a loop,
If two jobs are created, it gives one to the manager IS to place on the workpool and then executes the remaining IS
If only one job is created, it executes this job next.
If no new job is created, this IS requests a new job from the manager IS.

34. Additional Comments on MASC Quickhull
Convex Hull
1/1/ :28 AM
Additional Comments on MASC Quickhull
For one million points this algorithm would require lg n = 20.
Note that increasing the ISs by only 5 (to 25) would allow 33.5 million points to be processed.
Even if (lg n) ISs are available for this algorithm, the actual number of ISs would likely to still be less than lg n.
It would be inefficient to assume that every time a new task is created, an idle IS would be available to execute it.
However, this algorithm should also provide a speedup, even if only a small number k of ISs are available.
The complexity of the running time will still be O(lg n).
The actual running time could be up to k times faster than for one IS.
There will be some loss of efficiency due to IS interactions.
This is probably a more practical approach.